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Cohomological equation and cocycle rigidity of discrete parabolic actions
a. | The MITRE Corporation, McLean, VA 22102, USA |
b. | Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA |
We study the cohomological equation for discrete horocycle maps on $ {\rm SL}(2, \mathbb{R}) $ and $ {\rm SL}(2, \mathbb{R})\times {\rm SL}(2, \mathbb{R}) $ via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of $ {\rm SL}(2, \mathbb{R}) $. Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of $ \operatorname{\mathfrak s\mathfrak l}(2, \mathbb{R}) $, and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for $ {\rm SL}(2, \mathbb{R}) $ in which all cases of irreducible, unitary representations of $ {\rm SL}(2, \mathbb{R}) $ can be studied simultaneously.
Finally, our results combine with those of a very recent paper by the authors to give cohomology results for discrete parabolic actions in regular representations of some general classes of simple Lie groups, providing a fundamental step toward proving differential local rigidity of parabolic actions in this general setting.
References:
[1] |
H. Cartan and R. Takahashi, Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes, volume 115. Hermann Paris, 1961. |
[2] |
D. Damianovic and A. Katok,
Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic actions, Discrete Contin. Dynam. Syst, 13 (2005), 985-1005.
doi: 10.3934/dcds.2005.13.985. |
[3] |
D. Damjanovic and A. Katok,
Local rigidity of partially hyperbolic actions Ⅰ. KAM method and k actions on the torus, Annals of Mathematics, 172 (2010), 1805-1858.
doi: 10.4007/annals.2010.172.1805. |
[4] |
D. Damjanovic and A. Katok,
Local rigidity of homogeneous parabolic actions: Ⅰ. a model case, Journal of Modern Dynamics, 5 (2011), 203-235.
doi: 10.3934/jmd.2011.5.203. |
[5] |
D. Damjanovic and J. Tanis,
Cocycle rigidity and splitting for some discrete parabolic actions, Discrete and Continuous Dynamical Systems, 34 (2014), 5211-5227.
doi: 10.3934/dcds.2014.34.5211. |
[6] |
L. Flaminio and G. Forni,
Invariant distributions and time averages for horocycle flows, Duke Mathematical Journal, 119 (2003), 465-526.
doi: 10.1215/S0012-7094-03-11932-8. |
[7] |
L. Flaminio and G. Forni,
Equidistribution of nilflows and applications to theta sums, Ergodic Theory and Dynamical Systems, 26 (2006), 409-433.
doi: 10.1017/S014338570500060X. |
[8] |
L. Flaminio and G. Forni, On effective equidistribution for higher step nilflows, arXiv preprint, arXiv: 1407.3640, 2014. |
[9] |
L. Flaminio, G. Forni and J. Tanis,
Effective equidistribution of twisted horocycle flows and horocycle maps, Geometric and Functional Analysis, 26 (2016), 1359-1448.
doi: 10.1007/s00039-016-0385-4. |
[10] |
A. Katok and R. J. Spatzier,
Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions, Math. Res. Lett., 1 (1994), 193-202.
doi: 10.4310/MRL.1994.v1.n2.a7. |
[11] |
A. Katok and A. Kononenko,
Cocycles' stability for partially hyperbolic systems, Mathematical Research Letters, 3 (1996), 191-210.
doi: 10.4310/MRL.1996.v3.n2.a6. |
[12] |
A. Katok and R. J. Spatzier, First cohomology of anosov actions of higher rank abelian groups and applications to rigidity, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 79 (1994), 131–156. |
[13] |
A. Katok and R. J. Spatzier,
Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Proc. Steklov Inst. Math., 216 (1997), 287-314.
|
[14] |
G. A. Margulis,
Finitely-additive invariant measures on Euclidean spaces, Ergodic Theory and Dynamical Systems, 2 (1982), 383-396.
doi: 10.1017/S014338570000167X. |
[15] |
F. I. Mautner,
Unitary representations of locally compact groups II, Annals of Mathematics, 52 (1950), 528-556.
doi: 10.2307/1969431. |
[16] |
D. Mieczkowski,
The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory, J. Mod. Dyn, 1 (2007), 61-92.
doi: 10.3934/jmd.2007.1.61. |
[17] |
F. A. Ramirez,
Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups, Journal of Modern Dynamics, 3 (2009), 335-357.
doi: 10.3934/jmd.2009.3.335. |
[18] |
J. Tanis,
The cohomological equation and invariant distributions for horocycle maps, Ergodic Theory and Dynamical systems, 34 (2014), 299-340.
doi: 10.1017/etds.2012.125. |
[19] |
J. Tanis and Z. J. Wang,
Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher rank Lie groups, Geom. Funct. Anal., 25 (2015), 1956-2020.
doi: 10.1007/s00039-015-0351-6. |
[20] |
A. F. M. Ter Elst and D. W. Robinson,
Elliptic operators on Lie groups, Acta Applicandae Mathematicae, 44 (1996), 133-150.
doi: 10.1007/BF00116519. |
[21] |
Z. J. Wang,
Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups, Geometric and Functional Analysis, 25 (2015), 1956-2020.
doi: 10.1007/s00039-015-0351-6. |
[22] |
Z. J. Wang, Cocycle Rigidity of Partially Hyperbolic Actions, Cocycle rigidity of partially hyperbolic actions, 2017. |
[23] |
R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81. Birkhäuser Verlag, Basel, 1984.
doi: 10.1007/978-1-4684-9488-4. |
show all references
References:
[1] |
H. Cartan and R. Takahashi, Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes, volume 115. Hermann Paris, 1961. |
[2] |
D. Damianovic and A. Katok,
Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic actions, Discrete Contin. Dynam. Syst, 13 (2005), 985-1005.
doi: 10.3934/dcds.2005.13.985. |
[3] |
D. Damjanovic and A. Katok,
Local rigidity of partially hyperbolic actions Ⅰ. KAM method and k actions on the torus, Annals of Mathematics, 172 (2010), 1805-1858.
doi: 10.4007/annals.2010.172.1805. |
[4] |
D. Damjanovic and A. Katok,
Local rigidity of homogeneous parabolic actions: Ⅰ. a model case, Journal of Modern Dynamics, 5 (2011), 203-235.
doi: 10.3934/jmd.2011.5.203. |
[5] |
D. Damjanovic and J. Tanis,
Cocycle rigidity and splitting for some discrete parabolic actions, Discrete and Continuous Dynamical Systems, 34 (2014), 5211-5227.
doi: 10.3934/dcds.2014.34.5211. |
[6] |
L. Flaminio and G. Forni,
Invariant distributions and time averages for horocycle flows, Duke Mathematical Journal, 119 (2003), 465-526.
doi: 10.1215/S0012-7094-03-11932-8. |
[7] |
L. Flaminio and G. Forni,
Equidistribution of nilflows and applications to theta sums, Ergodic Theory and Dynamical Systems, 26 (2006), 409-433.
doi: 10.1017/S014338570500060X. |
[8] |
L. Flaminio and G. Forni, On effective equidistribution for higher step nilflows, arXiv preprint, arXiv: 1407.3640, 2014. |
[9] |
L. Flaminio, G. Forni and J. Tanis,
Effective equidistribution of twisted horocycle flows and horocycle maps, Geometric and Functional Analysis, 26 (2016), 1359-1448.
doi: 10.1007/s00039-016-0385-4. |
[10] |
A. Katok and R. J. Spatzier,
Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions, Math. Res. Lett., 1 (1994), 193-202.
doi: 10.4310/MRL.1994.v1.n2.a7. |
[11] |
A. Katok and A. Kononenko,
Cocycles' stability for partially hyperbolic systems, Mathematical Research Letters, 3 (1996), 191-210.
doi: 10.4310/MRL.1996.v3.n2.a6. |
[12] |
A. Katok and R. J. Spatzier, First cohomology of anosov actions of higher rank abelian groups and applications to rigidity, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 79 (1994), 131–156. |
[13] |
A. Katok and R. J. Spatzier,
Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Proc. Steklov Inst. Math., 216 (1997), 287-314.
|
[14] |
G. A. Margulis,
Finitely-additive invariant measures on Euclidean spaces, Ergodic Theory and Dynamical Systems, 2 (1982), 383-396.
doi: 10.1017/S014338570000167X. |
[15] |
F. I. Mautner,
Unitary representations of locally compact groups II, Annals of Mathematics, 52 (1950), 528-556.
doi: 10.2307/1969431. |
[16] |
D. Mieczkowski,
The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory, J. Mod. Dyn, 1 (2007), 61-92.
doi: 10.3934/jmd.2007.1.61. |
[17] |
F. A. Ramirez,
Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups, Journal of Modern Dynamics, 3 (2009), 335-357.
doi: 10.3934/jmd.2009.3.335. |
[18] |
J. Tanis,
The cohomological equation and invariant distributions for horocycle maps, Ergodic Theory and Dynamical systems, 34 (2014), 299-340.
doi: 10.1017/etds.2012.125. |
[19] |
J. Tanis and Z. J. Wang,
Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher rank Lie groups, Geom. Funct. Anal., 25 (2015), 1956-2020.
doi: 10.1007/s00039-015-0351-6. |
[20] |
A. F. M. Ter Elst and D. W. Robinson,
Elliptic operators on Lie groups, Acta Applicandae Mathematicae, 44 (1996), 133-150.
doi: 10.1007/BF00116519. |
[21] |
Z. J. Wang,
Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups, Geometric and Functional Analysis, 25 (2015), 1956-2020.
doi: 10.1007/s00039-015-0351-6. |
[22] |
Z. J. Wang, Cocycle Rigidity of Partially Hyperbolic Actions, Cocycle rigidity of partially hyperbolic actions, 2017. |
[23] |
R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81. Birkhäuser Verlag, Basel, 1984.
doi: 10.1007/978-1-4684-9488-4. |
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